The lambert W function is defined to be the inverse of $f(x)=xe^x$, and the equation $x^x=k$ can be solved fairly easily using the function: $$x^x=k$$ $$\ln(x^x)=\ln(k)$$ $$x\ln(x)=\ln(k)$$ $$e^{\ln(x)}\ln(x)=\ln(k)$$ $$\ln(x)=W(\ln(k))$$ $$x=e^{W(\ln(k))}$$
My question is, can $ x^{x^x}=k$ be solved using the lambert W function, and if so, can power towers of any height be solved using some iterative process?
Using Maurer's notation, the equation ${^n}x=y$ is solvable using Lambert's $W$ function only for $n=2$.
If we loosely define "transcendence degree of an equation" to be the height of the highest exponential tower with $x$ on top, contained in the equation minus one, then $W$ solves ONLY equations with (transcendence) degree at most 1 (that is, with exponentials of total height 2), but NOT all such equations.
The equation $x^{x^x}=y$ is written as ${^3}x=y$, consequently it has (transcendence) degree 2 (tower with 3 x's minus 1)
If memory serves right, the most general equation solvable by $W$ is something like:
$$a\cdot x^m\cdot d^{b\cdot x^n+c}=y$$
which as you can see can be reduced to having degree 1 at some point in the algebraic solution process, by taking $n$-th roots on both sides of the equation.
Elaborating on the second point, note that adding one plain $x$ to the equation above and it fails solvability. That is, the equation
$$a\cdot x^m\cdot d^{b\cdot x^n+c}+x=y$$
for example, is now NOT solvable by $W$, although it is of the same transcendence degree as the one above, so the definition of "transcendence degree" of an equation, needs to be made a liitle more rigorous, by specifying some additional configuration constraints, which vary from case to case.
Sure they can. Omitting details about existence, the following equivalence holds:
$${^n}x=y\Leftrightarrow x=^{\frac{1}{n}}y$$
and this is the $n$-th order super root of $y$, which, after taking care of domains and ranges, can be approximated using backwards numerical iteration.
Such a method, for example, was presented in Maple by Robert Israel, above, long time ago on sci.math.